A145388 -id:A145388 - cp's OEIS frontend

A384047 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is a unitary divisor of n.

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 2, 3, 2, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 3, 4, 1, 3, 1, 4, 3, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

Amiram Eldar, May 18 2025

			Triangle begins:
  1
  1, 2
  1, 1, 3
  1, 1, 1, 4
  1, 1, 1, 1, 5
  1, 2, 3, 2, 1, 6
  1, 1, 1, 1, 1, 1, 7
  1, 1, 1, 1, 1, 1, 1, 8
  1, 1, 1, 1, 1, 1, 1, 1, 9
  1, 2, 1, 2, 5, 2, 1, 2, 1, 10

Amiram Eldar, Table of n, a(n) for n = 1..10585 (first 145 rows flattened)

Cf. A005117, A050873, A077610, A145388 (row sums), A165430, A225174, A384046.

Upper right triangle of A322482.

udiv[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; T[n_, k_] := Max[Intersection[udiv[n], Divisors[k]]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten

udiv(n) = select(x -> gcd(x, n/x) == 1, divisors(n));
T(n, k) = vecmax(setintersect(udiv(n), divisors(k)));

T(n, 1) = 1.
T(n, n) = n.
T(n, k) <= A050873(n, k) = gcd(n, k), with equality if n is squarefree (A005117).

A065472 Decimal expansion of Product_{p prime} (1 - 1/(p+1)^2).

Original entry on oeis.org

7, 7, 5, 8, 8, 3, 5, 1, 0, 0, 0, 3, 8, 9, 5, 4, 9, 9, 6, 2, 0, 4, 0, 4, 2, 8, 4, 4, 2, 7, 9, 0, 0, 6, 1, 1, 4, 8, 2, 4, 1, 3, 4, 6, 5, 9, 7, 3, 0, 1, 6, 2, 7, 6, 2, 2, 1, 0, 6, 3, 1, 1, 6, 4, 6, 1, 3, 8, 7, 6, 4, 9, 2, 4, 9, 7, 4, 5, 6, 9, 9, 5, 3, 7, 1, 9, 3, 1, 3, 2, 3, 3, 1, 2, 8, 1, 4, 2

Offset: 0

N. J. A. Sloane, Nov 19 2001

The probablity that two randomly chosen squarefree numbers are coprime. - Amiram Eldar, Aug 04 2020

The asymptotic mean of A001157(n)/(n*A000203(n)). - Richard R. Forberg, May 27 2023

			0.7758835100038954996204042844279...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
László Tóth, The unitary analogue of Pillai's arithmetical function, Collectanea Mathematica, Vol. 40, No. 1 (1989), pp. 19-30.

Cf. A000203, A001157, A008683, A038063, A078091, A116393, A145388.

digits = 98; Exp[NSum[(-1)^n*(2^(n-1)-2)*PrimeZetaP[n-1]/(n-1), {n, 3, Infinity}, WorkingPrecision -> 2 digits, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p+1)^2) \\ Amiram Eldar, Mar 17 2021

Equals lim_{n->oo} (Pi^2/(3*n^2*log(n))) * Sum_{k=1..n} A145388(k). - Amiram Eldar, May 14 2019

Equals Sum_{k>=1} mu(k)/sigma(k)^2, where mu is the Möbius function (A008683) and sigma(k) is the sum of divisors of k (A000203). - Amiram Eldar, Jan 14 2022

Definition corrected by Dan Asimov, Apr 15 2006

A343525 If n = Product (p_j^k_j) then a(n) = Product (2*p_j^k_j + 1), with a(1) = 1.

Original entry on oeis.org

1, 5, 7, 9, 11, 35, 15, 17, 19, 55, 23, 63, 27, 75, 77, 33, 35, 95, 39, 99, 105, 115, 47, 119, 51, 135, 55, 135, 59, 385, 63, 65, 161, 175, 165, 171, 75, 195, 189, 187, 83, 525, 87, 207, 209, 235, 95, 231, 99, 255, 245, 243, 107, 275, 253, 255, 273, 295, 119, 693, 123, 315, 285, 129, 297, 805

Offset: 1

Ilya Gutkovskiy, Apr 18 2021

The unitary analog of A060640.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A001221, A034444, A034448, A060640, A107759, A145388, A298473.

a:= n-> mul(2*i[1]^i[2]+1, i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 18 2021

a[1] = 1; a[n_] := Times @@ ((2 #[[1]]^#[[2]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 66}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2*f[k,1]^f[k,2]+1; f[k,2]=1); factorback(f); \\ Michel Marcus, Apr 18 2021

a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * usigma(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * 2^omega(d).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-1) - 2/p^(2*s-1)). - Amiram Eldar, Jul 24 2024

A333557 a(n) = Sum_{d|n, gcd(d, n/d) = 1} uphi(d) * uphi(n/d), where uphi = unitary totient function (A047994).

Original entry on oeis.org

1, 2, 4, 6, 8, 8, 12, 14, 16, 16, 20, 24, 24, 24, 32, 30, 32, 32, 36, 48, 48, 40, 44, 56, 48, 48, 52, 72, 56, 64, 60, 62, 80, 64, 96, 96, 72, 72, 96, 112, 80, 96, 84, 120, 128, 88, 92, 120, 96, 96, 128, 144, 104, 104, 160, 168, 144, 112, 116, 192, 120, 120, 192, 126, 192, 160, 132, 192, 176, 192

Offset: 1

Ilya Gutkovskiy, Mar 26 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A029935, A034444, A047994, A055653, A062355, A145388.

uphi[1] = 1; uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[n_] := Sum[If[GCD[d, n/d] == 1, uphi[d] uphi[n/d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
Table[Sum[If[GCD[d, n/d] == 1, (-2)^PrimeNu[n/d] 2^PrimeNu[d] d, 0], {d, Divisors[n]}], {n, 1, 70}]
f[p_, e_] := 2*(p^e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

a(n) = sumdiv(n, d, if (gcd(d, n/d) == 1, (-2)^omega(n/d)*2^omega(d)*d)); \\ Michel Marcus, Mar 27 2020

If n = Product (p_j^k_j) then a(n) = Product (2 * (p_j^k_j - 1)).
a(n) = 2^omega(n) * uphi(n).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-2)^omega(n/d) * 2^omega(d) * d.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * A145388(d).

A333576 a(1) = 1; thereafter a(n) = n * uphi(n) / 2.

Original entry on oeis.org

1, 1, 3, 6, 10, 6, 21, 28, 36, 20, 55, 36, 78, 42, 60, 120, 136, 72, 171, 120, 126, 110, 253, 168, 300, 156, 351, 252, 406, 120, 465, 496, 330, 272, 420, 432, 666, 342, 468, 560, 820, 252, 903, 660, 720, 506, 1081, 720, 1176, 600, 816, 936, 1378, 702, 1100, 1176, 1026, 812, 1711, 720

Offset: 1

Ilya Gutkovskiy, Mar 27 2020

The unitary version of A023896.

Cf. A001221, A023896, A047994, A065464, A076479, A145388, A353908.

uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[1] = 1; a[n_] := n uphi[n]/2; Table[a[n], {n, 1, 60}]
a[n_] := (n/2) Sum[If[GCD[d, n/d] == 1, (-1)^PrimeNu[n/d] (d + 1), 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

a(n) = if(n == 1, 1, my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] - 1) * n / 2); \\ Amiram Eldar, Sep 21 2024

a(n) = (n/2) * Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * (d + 1).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/36) * Product_{p prime} (1 - (2*p-1)/p^3) = A353908 * A065464 = 0.117407... . - Amiram Eldar, Sep 21 2024

A380650 The largest number which is a linear combination of the divisors of n with nonnegative integer coefficients such that no linear combination with smaller nonnegative integer coefficients is equal to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 7, 8, 13, 10, 17, 12, 19, 22, 15, 16, 25, 18, 31, 32, 31, 22, 37, 24, 37, 26, 45, 28, 60, 30, 31, 52, 49, 58, 59, 36, 55, 62, 67, 40, 85, 42, 73, 76, 67, 46, 77, 48, 73, 82, 87, 52, 79, 94, 97, 92, 85, 58

Offset: 1

Alexei Vernitski, Jan 29 2025

nonn

The mean of this sequence and Euler's totient function A000010 is approximately (but not exactly) equal to n.
The definition has evolved from a recreational question asked by P. M. Higgins, asking what maximal sum of money can be produced using British coins so no sum of one pound is produced by any subset of these coins.
The terms up to and including a(29)=28 agree with the formula a(n) = (A145388(n) - 1)/2, but a(30)=60, while the formula gives 67. This difference should be confirmed by an independent calculation using the definition in the name. - Hugo Pfoertner, Feb 14 2025

			For n = 12, the largest sum is 17 = 0*1 + 0*2 + 1*3 + 2*4 + 1*6 = 0*1 + 0*2 + 3*3 + 2*4 + 0*6.
For n = 30, the largest sum is 60 = 1*1 + 0*2 + 0*3 + 0*5 + 4*6 + 2*10 + 1*15.

Cf. A000010, A145388.

cp's OEIS Frontend

A384047 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is a unitary divisor of n.

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A065472 Decimal expansion of Product_{p prime} (1 - 1/(p+1)^2).

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A343525 If n = Product (p_j^k_j) then a(n) = Product (2*p_j^k_j + 1), with a(1) = 1.

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A333557 a(n) = Sum_{d|n, gcd(d, n/d) = 1} uphi(d) * uphi(n/d), where uphi = unitary totient function (A047994).

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A333576 a(1) = 1; thereafter a(n) = n * uphi(n) / 2.

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A380650 The largest number which is a linear combination of the divisors of n with nonnegative integer coefficients such that no linear combination with smaller nonnegative integer coefficients is equal to n.

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